let C be concrete category; :: thesis: for a, b being object of C st <^a,b^> <> {} & <^b,a^> <> {} holds
for f being Morphism of a,b st f is retraction holds
rng f = the_carrier_of b
let a, b be object of C; :: thesis: ( <^a,b^> <> {} & <^b,a^> <> {} implies for f being Morphism of a,b st f is retraction holds
rng f = the_carrier_of b )
assume A1: ( <^a,b^> <> {} & <^b,a^> <> {} ) ; :: thesis: for f being Morphism of a,b st f is retraction holds
rng f = the_carrier_of b
let f be Morphism of a,b; :: thesis: ( f is retraction implies rng f = the_carrier_of b )
given g being Morphism of b,a such that A2: g is_right_inverse_of f ; :: according to ALTCAT_3:def 2 :: thesis: rng f = the_carrier_of b
A3: f * g = idm b by A2, ALTCAT_3:def 1;
A4: f * g = f * g by A1, Th38;
A5: ( f is Function of (the_carrier_of a),(the_carrier_of b) & g is Function of (the_carrier_of b),(the_carrier_of a) ) by A1, Th35;
idm b = id (the_carrier_of b) by Def10;
hence rng f = the_carrier_of b by A3, A4, A5, FUNCT_2:24; :: thesis: verum